0.2 Laplace transform of sinc function

By the formula ℒ⁢{f′}=s⁢ℒ⁢{f}-limx→0+⁡f⁢(x)ℒsuperscriptfnormal-′sℒfsubscriptnormal-→xlimit-from0fx\mathcal{L}\{f^{{\prime}}\}=s\mathcal{L}\{f\}-\lim_{{x\to 0+}}f(x) of the parent entry,
we obtain as consequence of (1), that

Now we have the derivative φ′⁢(a)=∫0∞e-s⁢t⁢cos⁡a⁢t⁢d⁢tsuperscriptφnormal-′asuperscriptsubscript0superscriptestatdt\varphi^{{\prime}}(a)=\int_{0}^{\infty}e^{{-st}}\cos{at}\,dt, where one can partially integrate twice, getting